Open-loop flux observers are used to estimate rotor position in an electric machine such as a permanent magnet (PM) motor. The open-loop flux observer is typically called a “sensorless” estimator because the rotor position is inferred rather than measured directly. Direct rotor position sensors typically include rotor position transducers (RPTs) or other sensors that sense movement of the rotor. Direct rotor position sensors are typically costly to implement and may tend to reduce the reliability of the electric machine.
The open-loop flux observer estimates rotor position using stator currents and commanded stator voltages as inputs. The open-loop flux observer calculates the back EMF of the electric machine. There are several characteristic equations that are used:                               Ψ          dqs          s                =                  ∫                                    (                                                V                  dqs                                      s                    *                                                  -                                                      R                    s                                    ·                                      i                    dqs                    s                                                              )                        ⁢                          ⅆ              t                                                          (        1        )                                          θ                      Ψ            ⁢                                                   ⁢            s                          =                              tan                          -              1                                ⁡                      (                                          Ψ                qs                s                                            Ψ                ds                s                                      )                                              (        2        )                                δ        =                                            tan                              -                1                                      ⁡                          (                                                Ψ                  qs                  e                                                  Ψ                  ds                  e                                            )                                =                                    tan                              -                1                                      ⁡                          (                                                                    L                    q                                    ·                                      i                    qs                    e                                                                                        Ψ                    f                                    +                                                            L                      d                                        ·                                          i                      ds                      e                                                                                  )                                                          (        3        )                                          θ          r                =                              θ                          Ψ              ⁢                                                           ⁢              s                                -          δ                                    (        4        )            Where Ψdqss is the stator flux linkage in the stationary reference frame, idqss is the stator current in the stationary reference frame, Rs is the stator resistance, Ψdss and Ψqss are the d-axis and q-axis stator flux linkages in the stationary reference frame, Ψdse and Ψqse are the d-axis and q-axis stator flux linkages in the stationary reference frame, Ψf is the permanent magnet flux linkage, Ld is the d-axis inductance, Lq is the q-axis inductance, idse and iqse are synchronous reference frame currents, θ, is the rotor position, θΨs is the angular position of the stator flux and δ is the load angular position.
The back EMF is integrated to obtain the stator flux linkage in a stationary reference frame (See Equation 1). The angular position of the stator flux is usually obtained using the arctangent function (See Equation 2). The rotor position information is obtained by subtracting the load angular position δ (See Equation 3) from the stator flux position (See Equation 4).
In most implementations, however, an integration function that is set forth in Equation 1 is not used. Cascaded low pass filters (LPF) are typically used to simulate the integration function to avoid integration problems that occur at low stator frequencies. Cascaded LPFs also provide improved transient response as compared to a single LPF since faster time constants can be used.
The conventional open-loop flux observer has several performance problems. The cascaded LPFs require electrical speed data, which is not normally available from basic open-loop observers. The electrical speed data is used to set the LPF coefficients. To generate the electrical speed data, a derivative of angular position is generated. The derivative operation tends to be noise sensitive and can create errors in the electrical speed data. The electrical speed data is used to compute the coefficients of the cascade LPFs. Errors in the electrical speed data adversely impact LPF characteristics such as gain and phase and may cause instability. Also, the conventional open-loop flux observer requires an arctangent function, which can be computationally intensive.